《金碧輝煌的聖殿 （3. Johnson’s theorem）》The Glorious Temple of Geometry

金碧輝煌的聖殿 （3. Johnson’s theorem）The Glorious Temple of Geometry (3.）

 

掛在我家牆上的幾何圖案中，有一個是在正方形的鏡框裏麵。這個幾何圖形包含四個圓，（我認為）極其優美。它顯示的是Johnson’s theorem。

Among the geometric patterns hanging on the walls of my home, there is one within a square frame. This geometric figure, containing four circles, is exceptionally beautiful, displaying Johnson's Theorem.

《幾何原本》寫於公元前4世紀，此後的幾何規律和定理又被一代代數學家和愛好者不斷發現和證明。一些有相當難度、很隱蔽的規律，比如下圖所示的“歐拉線定理”，也被歐拉這樣的天才數學家在260年前發現並證明了—

The Elements was written in the 4th century BC, and since then, geometric laws and theorems have been continually discovered and proven by generations of mathematicians and enthusiasts. Some challenging and obscure rules, like the "Euler Line Theorem" shown in the figure below, were discovered and proven by geniuses like Euler 260 years ago --

然而這座“金礦”似乎總有開采不完的金子。上一篇我們聊了Morley’s trisector theorem，它是在19世紀的最後一年，即1899年被揭示的。那麽，到了20世紀，還能不能有那樣簡明而優美的發現呢？沒問題！

However, this "gold mine" seems to always yield inexhaustible treasures. As discussed in my previous article, we talked about Morley’s trisector theorem, revealed in the last year of the 19th century, 1899. So, can the 20th century bring forth discoveries as concise and elegant? Absolutely!

Johnson’s Theorem 發表於1916年，已經是20世紀了。它是這樣表述的：“如果三個半徑同為r的圓經過一公共點H，那麽經過另外三個交點的圓的半徑也是r 【Given three circles of equal radii r that all pass through a common point H, then the circle through their other three intersections has the same radius r.】換句話說，以另外三個交點為頂角的三角形，其外接圓與原先那三個圓是全等的。簡單不簡單？

Johnson’s Theorem was published in 1916, firmly placing it in the 20th century. It is stated as follows: "Given three circles of equal radii r that all pass through a common point H, then the circle through their other three intersections has the same radius r." In other words, the circumcircle of the triangle formed by the other three intersection points is congruent to the original three circles. Simple, isn't it?

發表這個定理的，是美國數學家Roger Johnson。他1913年獲得哈佛大學數學博士學位，不久以後進入紐約的Hunter College數學係。他後來長期擔任係主任直到退休。Roger Johnson不是大數學家，他一生僅發表十幾篇論文（包括上圖中所示的僅一頁的關於該定理的論文 A circle theorem。）但他的一部探討歐式幾何複雜問題的專著Modern Geometry - An Elementary Treatise on the Geometry of the Triangle and the Circle (Houghton Mifflin, 1929) 卻在幾十年裏頗歡迎。現在這本書的全文都可以在網上閱讀。

The mathematician who published this theorem was the American Roger Johnson. He earned his Ph.D. in mathematics from Harvard University in 1913 and soon after joined the mathematics department at Hunter College in New York. He later served as the department chair until his retirement. Johnson was not a prominent mathematician, and only published just over a dozen papers in his lifetime (including the one-page paper on this theorem shown in the figure). Nevertheless, his book Modern Geometry - An Elementary Treatise on the Geometry of the Triangle and the Circle (Houghton Mifflin, 1929) was well-received for decades and is now available for online reading.

這篇小論文最後一段的討論，其口氣很有意思：我這發現啊，“appear to be new”，可是它如此簡單，明確，兩千多年來難道所有的人都忽視了？我有點兒心虛啊。讀者們，你們如果發現我不是原創，請報告…… 這種論文的寫法今天肯定是沒有的。

The concluding paragraph of this short paper has an interesting tone: My discovery, "appears to be new," but it is so simple and clear. Could everyone have overlooked it for more than two thousand years? I feel a bit uneasy. Readers, if you find that I am not the originator, please report... Such a writing style is obsolete in academic papers today.

上麵兩個圖看起來有些深奧，其實隻是前麵那個圖稍加擴展。圖A顯示了第五個圓，它是以三個圓的交點H為圓心，作一個圓，經過另外三個圓的圓心，顯然這個圓的半徑也是r，所以這個圖中五個圓的半徑都相同。圖B我不細解釋了。有一點幾何童子功的網友馬上就能看出來，那個紅色大圓的半徑為2r。那三個半徑為r的圓無論怎樣移動，隻要有共點H，它們始終都是與大圓內切的。

The two figures above may seem profound, but they are just a slight extension of the first figure. Figure A shows the fifth circle, which is centered at the intersection point H of three circles, passing through the centers of the other three circles. Clearly, the radius of this circle is also r, so the radii of all five circles in this figure are the same. I won't explain Figure B in detail. Geometry enthusiasts can quickly see that the radius of the large red circle is 2r. The three circles with a radius of r, no matter how they move, as long as they have a common point H, they are always tangent to the large circle.

 

上麵這個圖，進一步表現這個幾何結構的特點：無論是把六個相關的點連成兩個中心對稱的全等三角形，還是連成如立方體的透視圖，深層次的規律是“萬變不離其宗”的。以橙紅色代表的Johnson環的半徑為r是不變的。

The above figure further illustrates the characteristics of this geometric structure: whether connecting the six related points into two centrally symmetric congruent triangles or connecting them into a perspective view of a cube, the deep-seated rule is "varied but not deviating from its essence." The radius of the Johnson circle, represented in orange-red, remains unchanged.

說到這裏，故事並沒有完...... Johnson的論文發表後又過了很多年，人們才發現，這個規律其實早在之前八年，即1908年，已經被一個叫Gheorghe Titeica的羅馬尼亞人發現並證明了。我本人在知道這件事以後，一開始以為這位Titeica先生是羅馬尼亞的某個偏遠地區的小鎮做題家，沒有話語權，讓美國佬Johnson占了便宜。

At this point, it seems like it should be the end. However, many years after Johnson published his paper, it was discovered that this rule had already been found and proven by a Romanian named Gheorghe Titeica eight years earlier, in 1908. Initially, I thought Mr. Titeica was a problem solver in a remote town in Romania, with no say, letting the American Johnson take advantage.

我完全錯了。Titeica是羅馬尼亞著名數學家。他在法國獲得博士學位（法國的數學在18-19世紀是世界上最牛的），他是羅馬尼亞微分幾何（differential geometry）的奠基人，他在羅馬尼亞國家科學院長期擔任高職，他是羅馬尼亞數學學會的主席。作為知名學者，他還是國際數學大會（ICM）幾何分支的主席。他發表過幾百篇論文，學生中也出了著名數學家，他兒子是知名量子物理學家，而且他本人還是小提琴高手，他牛得很。

I was completely wrong. Titeica was a renowned Romanian mathematician. He earned his Ph.D. in France (French mathematics was the world's best in the 18th and 19th centuries), and he laid the foundation for differential geometry in Romania. He held high positions in the Romanian Academy of Sciences for a long time and served as the president of the Romanian Mathematical Society. As a well-known scholar, he also served as the president of the geometry branch of the International Congress of Mathematicians (ICM). He published hundreds of papers, produced famous mathematicians among his students, and his son was a well-known quantum physicist. Moreover, he was a skilled violinist. He was truly remarkable.

有一天他拿著一枚5-lei 硬幣，畫了3個共點的圓，突然意識到還有一個全等的圓隱含在裏麵，他畫了出來。至於證明，對他來說實在是小菜一碟。這個平麵幾何的小問題，簡單又直觀，他根本不認為是什麽大的發現。恰巧這時候羅馬尼亞的《數學學報》（Gazeta Matematica）有一個開放性的數學競賽，讓參賽者自己提出命題並證明。當時在布加勒斯特大學當教授的Titeica，就把自己剛剛發現的“the five-lei coin problem”加上證明，用羅馬尼亞語撰寫後遞交上去了。那是1908年。

One day, holding a five-lei coin, he drew three circles with a common point and suddenly realized that another congruent circles were implicit in them, so he drew it. As for the proof, it was a piece of cake for him. This small problem in plane geometry was simple and straightforward, and he didn't consider it a significant discovery. Coincidentally, at that time, there was an open mathematical competition in the Romanian Gazeta Matematica, allowing participants to propose and prove their propositions. Titeica, who was a professor at the University of Bucharest at the time, submitted this "the five-lei coin problem" with proof, written in Romanian.  That was in year 1908.

Titeica教授在數學方麵的貢獻，根本不以這個“小兒科”發現說事兒。1961年羅馬尼亞為他專門發行了一枚郵票，是表彰他在微分幾何方麵的成就和他的leadership。Titeica也不爭什麽名分，倒是數學愛好者們常有不平。Roger Johnson雖然是獨立發現了相同的規律，但Gheorghe Titeica畢竟比他早8年，也是（用非英語）記錄在案的。若稱該定理為Titeica-Johnson’s Theorem 應該更公平。我同意這一點。

Professor Titeica's contribution to mathematics does not hinge on this small discovery. In 1961, Romania issued a postage stamp specifically to honor his achievements in differential geometry and his leadership. Titeica did not seek fame on finding and proving the circle problem, but mathematics enthusiasts often express dissatisfaction. Although Roger Johnson independently discovered the same rule, Gheorghe Titeica was eight years ahead, and it was clearly recorded (in a non-English language). Calling the theorem Titeica-Johnson’s Theorem would be fairer. I agree with this.

到了1999年（Titeica已經去世整整60年了），第40屆“國際數學奧林匹克”（IMO）在羅馬尼亞舉行。那神秘的第四個圓再次靜悄悄地浮出水麵，嵌入大會的Logo（見上）。我認為這個極好的設計 — 第一，第四個圓正好用到“40”裏；第二，這是羅馬尼亞學者首先發現的；第三，這是屬於基礎數學範疇，與IMO正相配；第四，定理簡明、圖形優美、Logo設計相當美觀！

In 1999 (60 years after Titeica's death), the 40th International Mathematical Olympiad (IMO) was held in Romania. The mysterious fourth circle surfaced quietly again, embedded in the logo of the event (see above). I think this design is excellent—first, the forth circle fits perfectly into "40"; second, it initially discovered by a Romanian scholar; third, it is on basic mathematics, fitting well with the IMO; fourth, the theorem is concise, the graphic is beautiful, and the logo design is elegant!

【edited from ChatGPT translation】